Multi-quadratic p-rational number fields

Benmerieme, Y.; Movahhedi, Abbas C.

doi:10.1016/j.jpaa.2020.106657

Cited by 9 publications

(13 citation statements)

References 26 publications

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“…Towards the end of [4], the totally real bi-quadratic field K α := Q( αp(αp + 2), αp(αp − 2)) has been mentioned, where p > 3 is a prime number and α is a postive integer with gcd(α, p) = 1. It has been explicitly written (at page number 15 of [4]) that "... the fundamental unit of each subfield of K α is not locally a p th power at the p-adic places.…”

Section: Discussionmentioning

confidence: 99%

“…Similarly, we let F j := Q( −p(p − j)). Then the discriminant d F j of F j satisfies the inequality 4 . Therefore, by using Proposition 2.5, we obtain (4) h…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Over the past few years, mathematicians have addressed Conjecture 1.1 for small values of t. In [3], Barbulescu and Ray proved that the quadratic field Q( p 2 − 1) is p-rational for all prime numbers p. In [4], Benmerieme and Movahhedi proved that Q( p(p + 2)), Q( p(p − 2)) and the bi-quadratic field Q( p(p + 2), p(p − 2)) are all p-rational for all prime p ≥ 5. These results confirm Conjecture 1.1 for all odd primes p and for t = 1 and 2.…”

Section: Introductionmentioning

confidence: 99%

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On the $p$-rationality of consecutive quadratic fields

Chattopadhyay¹,

Laxmi²,

Saikia³

2022

Preprint

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In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic p-rational number fields of degree 2 t for any odd prime number p and any integer t ≥ 1. Using the criteria provided by him to check p-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic p-rational fields have been shown to exist in recent years. In this article, for any integer k ≥ 1, we build upon the existing work and prove the existence of infinitely many prime numbers p for which the imaginary quadratic fields Q( −(p − 1)), . . . , Q( −(p − k)) and Q( −p(p − 1)), . . . , Q( −p(p − k)) are all p-rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of p-rationality for two consecutive real quadratic fields by proving the existence of infinitely many p-rational fields of the form Q( p 2 + 1) and Q( p 2 + 2). The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the density of square-free values of polynomials at prime arguments.

show abstract

Section: Discussionmentioning

confidence: 99%

“…Similarly, we let F j := Q( −p(p − j)). Then the discriminant d F j of F j satisfies the inequality 4 . Therefore, by using Proposition 2.5, we obtain (4) h…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the $p$-rationality of consecutive quadratic fields

Chattopadhyay¹,

Laxmi²,

Saikia³

2022

Preprint

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“…Remark 2.2. The assumption on p only guarantees that µ p ⊂ K v for each place v of K dividing p. The previous proposition is proved in two different ways in [10] and [3]. In [3], the authors also give alternative criteria of p-rationality for real quadratic fields.…”

Section: Criteria Of P-rationality For Quadratic Numbers Fieldsmentioning

confidence: 99%

“…[3] Prop.4.4). For all prime p ≥ 5, the real quadratic fields Q( p(p − 2)), Q( p(p + 2)) and Q( (p − 2)(p + 2)) are p-rational.…”

mentioning

confidence: 97%

Triquadratic p-Rational Fields

Koperecz¹

2021

Preprint

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In his work about Galois representations, Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2 t . In this article, we prove that there exists infinitely many primes p such that the triquadratic fieldTo do this, we use an analytic result, proved apart in section §4, providing us with infinitely many prime numbers p such that p + 2 et p − 2 have "big" square factors. Therefore the related imaginary quadratic subfieldshave "small" discriminants for infinitely many primes p. In the spirit of Brauer-Siegel estimates, it proves that the class numbers of these imaginary quadratic fields are relatively prime to p, and so prove their p-rationality.

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On p-rationality of Q(ζ2l+1)+<

Lim

2022

Journal of Number Theory

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Multi-quadratic p-rational number fields

Cited by 9 publications

References 26 publications

On the $p$-rationality of consecutive quadratic fields

On the $p$-rationality of consecutive quadratic fields

Triquadratic p-Rational Fields

On p-rationality of Q(ζ2l+1)+<

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